Theorem rmoeqd 2705

Description: Equality deduction for restricted at-most-one quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.)

Hypothesis

Ref	Expression
raleqd.1	|- ( A = B -> ( ph <-> ps ) )

Assertion

Ref	Expression
rmoeqd	|- ( A = B -> ( E* x e. A ph <-> E* x e. B ps ) )

Distinct variable groups:   x, A   x, B

Allowed substitution hints:   ph( x)   ps( x)

Proof of Theorem rmoeqd

Step	Hyp	Ref	Expression
1		rmoeq1 2693	. 2 |- ( A = B -> ( E* x e. A ph <-> E* x e. B ph ) )
2		raleqd.1	. . 3 |- ( A = B -> ( ph <-> ps ) )
3	2	rmobidv 2683	. 2 |- ( A = B -> ( E* x e. B ph <-> E* x e. B ps ) )
4	1, 3	bitrd 188	1 |- ( A = B -> ( E* x e. A ph <-> E* x e. B ps ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   E*wrmo 2475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rmo 2480

This theorem is referenced by: (None)